CS 224n Assignment #2: word2vec (43 Points)

• Written: Understanding word2vec (23 points)

Let’s have a quick refresher on the word2vec algorithm. The key insight behind word2vec is that ‘a word is known by the company it keeps’. Concretely, suppose we have a ‘center’ word c and a contextual window surrounding c. We shall refer to words that lie in this contextual window as ‘outside words’. For example, in Figure 1 we see that the center word c is ‘banking’. Since the context window size is 2, the outside words are ‘turning’, ‘into’, ‘crises’, and ‘as’.

The goal of the skip-gram word2vec algorithm is to accurately learn the probability distribution P (OjC). Given a speci c word o and a speci c word c, we want to calculate P (O = ojC = c), which is the probability that word o is an ‘outside’ word for c, i.e., the probability that o falls within the contextual window of c.

Figure 1: The word2vec skip-gram prediction model with window size 2

In word2vec, the conditional probability distribution is given by taking vector dot-products and applying the softmax function:

exp(u>vc)

o (1)

exp(u>vc)

w2Vocab w

Here, uo is the ‘outside’ vector representing outside word o, and vc is the ‘center’ vector representing center word c. To contain these parameters, we have two matrices, U and V . The columns of U are all the ‘outside’ vectors uw. The columns of V are all of the ‘center’ vectors vw. Both U and V contain a vector for every w 2 Vocabulary.1

Recall from lectures that, for a single pair of words c and o, the loss is given by:

Jnaive-softmax(vc; o; U) = log P (O = ojC = c):

(2)

Another way to view this loss is as the cross-entropy2 between the true distribution y and the predicted distribution y^. Here, both y and y^ are vectors with length equal to the number of words in the vocabulary. Furthermore, the kth entry in these vectors indicates the conditional probability of the kth word being an ‘outside word’ for the given c. The true empirical distribution y is a one-hot vector with a 1 for the true out-side word o, and 0 everywhere else. The predicted distribution y^ is the probability distribution P (OjC = c) given by our model in equation (1).

• Assume that every word in our vocabulary is matched to an integer number k. Bolded lowercase letters represent vectors. uk is both the kth column of U and the ‘outside’ word vector for the word indexed by k. vk is both the kth column of V and the ‘center’ word vector for the word indexed by k. In order to simplify notation we shall interchangeably use k to refer to the word and the index-of-the-word.

2The Cross Entropy Loss between the true (discrete) probability distribution p and another distribution q is P

1

CS 224n Assignment #2: word2vec (43 Points)

(a) (3 points) Show that the naive-softmax loss given in Equation (2) is the same as the cross-entropy loss between y and y^; i.e., show that

X

yw log(^yw) = log(^yo):

(3)

w2V ocab

Your answer should be one line.

(b) (5 points) Compute the partial derivative of Jnaive-softmax(vc; o; U) with respect to vc. Please write your answer in terms of y, y^, and U.

(c) (5 points) Compute the partial derivatives of Jnaive-softmax(vc; o; U) with respect to each of the ‘outside’ word vectors, uw’s. There will be two cases: when w = o, the true ‘outside’ word vector, and w 6= o, for all other words. Please write you answer in terms of y, y^, and vc.

(d) (3 Points) The sigmoid function is given by Equation 4:

1

ex

(x) =

=

(4)

x

x

+ 1

1 + e

e

Please compute the derivative of (x) with respect to x, where x is a scalar. Hint: you may want to write your answer in terms of (x).

(e) (4 points) Now we shall consider the Negative Sampling loss, which is an alternative to the Naive

Softmax loss. Assume that K negative samples (words) are drawn from the vocabulary. For simplicity of notation we shall refer to them as w1; w2; : : : ; wK and their outside vectors as u1; : : : ; uK . Note that o 2= fw1; : : : ; wK g. For a center word c and an outside word o, the negative sampling loss function is given by:

K

Jneg-sample(vc; o; U) = log( (uo>vc))

kX

log( ( uk>vc))

(5)

=1

for a sample w1; : : : wK , where ( ) is the sigmoid function.3

Please repeat parts (b) and (c), computing the partial derivatives of Jneg-sample with respect to vc, with respect to uo, and with respect to a negative sample uk. Please write your answers in terms of the vectors uo, vc, and uk, where k 2 [1; K]. After you’ve done this, describe with one sentence why this loss function is much more e cient to compute than the naive-softmax loss. Note, you should be able to use your solution to part (d) to help compute the necessary gradients here.

(f) (3 points) Suppose the center word is c = wt and the context window is [wt m, : : :, wt 1, wt, wt+1, : : :, wt+m], where m is the context window size. Recall that for the skip-gram version of word2vec, the total loss for the context window is:

X

Jskip-gram(vc; wt m; : : : wt+m; U) = J(vc; wt+j; U) (6)

m j m

j=06

Here, J(vc; wt+j; U) represents an arbitrary loss term for the center word c = wt and outside word wt+j. J(vc; wt+j; U) could be Jnaive-softmax(vc; wt+j; U) or Jneg-sample(vc; wt+j; U), depending on your implementation.

Write down three partial derivatives:

• Note: the loss function here is the negative of what Mikolov et al. had in their original paper, because we are doing a minimization instead of maximization in our assignment code. Ultimately, this is the same objective function.

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CS 224n Assignment #2: word2vec (43 Points)

(i) @Jskip-gram(vc; wt m; : : : wt+m; U)=@U

(ii) @Jskip-gram(vc; wt m; : : : wt+m; U)=@vc

(iii) @Jskip-gram(vc; wt m; : : : wt+m; U)=@vw when w 6= c

Write your answers in terms of @J(vc; wt+j; U)=@U and @J(vc; wt+j; U)=@vc. This is very simple { each solution should be one line.

Once you’re done: Given that you computed the derivatives of J(vc; wt+j; U) with respect to all the model parameters U and V in parts (a) to (c), you have now computed the derivatives of the full loss function Jskip-gram with respect to all parameters. You’re ready to implement word2vec!

• Coding: Implementing word2vec (20 points)

In this part you will implement the word2vec model and train your own word vectors with stochastic gradient descent (SGD). Before you begin, rst run the following commands within the assignment directory in order to create the appropriate conda virtual environment. This guarantees that you have all the necessary packages to complete the assignment. Also note that you probably want to nish the previous math section before writing the code since you will be asked to implement the math functions in Python. You want to implement and test the following subsections in order since they are accumulative.

conda env create -f env.yml

conda activate a2

Once you are done with the assignment you can deactivate this environment by running:

conda deactivate

For each of the methods you need to implement, we included approximately how many lines of code our solution has in the code comments. These numbers are included to guide you. You don’t have to stick to them, you can write shorter or longer code as you wish. If you think your implementation is signi cantly longer than ours, it is a signal that there are some numpy methods you could utilize to make your code both shorter and faster. for loops in Python take a long time to complete when used over large arrays, so we expect you to utilize numpy methods. We will be checking the e ciency of your code. You will be able to see the results of the autograder when you submit your code to Gradescope.

(a) (12 points) We will start by implementing methods in word2vec.py. First, implement the sigmoid method, which takes in a vector and applies the sigmoid function to it. Then implement the softmax loss and gradient in the naiveSoftmaxLossAndGradient method, and negative sampling loss and gradient in the negSamplingLossAndGradient method. Finally, ll in the implementation for the skip-gram model in the skipgram method. When you are done, test your implementation by running python word2vec.py.

(b) (4 points) Complete the implementation for your SGD optimizer in the sgd method of sgd.py. Test your implementation by running python sgd.py.

(c) (4 points) Show time! Now we are going to load some real data and train word vectors with everything you just implemented! We are going to use the Stanford Sentiment Treebank (SST) dataset to train word vectors, and later apply them to a simple sentiment analysis task. You will need to fetch the datasets rst. To do this, run sh get datasets.sh. There is no additional code to write for this part; just run python run.py.

Note: The training process may take a long time depending on the e ciency of your implementation and the compute power of your machine (an e cient implementation takes one to two hours). Plan accordingly!

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CS 224n Assignment #2: word2vec (43 Points)

After 40,000 iterations, the script will nish and a visualization for your word vectors will appear. It will also be saved as word vectors.png in your project directory. Include the plot in your homework write up. Brie y explain in at most three sentences what you see in the plot.

• Submission Instructions

You shall submit this assignment on GradeScope as two submissions { one for \Assignment 2 [coding]"

and another for ‘Assignment 2 [written]":

(a) Run the collect submission.sh script to produce your assignment2.zip le.

(b) Upload your assignment2.zip le to GradeScope to \Assignment 2 [coding]".

(c) Upload your written solutions to GradeScope to \Assignment 2 [written]".

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